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Abstract

The aim of this work was to design and control, using genetic algorithm (GA) for parameter
optimization, one-charge-qubit quantum logic gates σx, σy, and σz, using two bound states as a qubit space, of circular graphene quantum dots in a
homogeneous magnetic field. The method employed for the proposed gate implementation
is through the quantum dynamic control of the qubit subspace with an oscillating electric
field and an onsite (inside the quantum dot) gate voltage pulse with amplitude and
time width modulation which introduce relative phases and transitions between states.
Our results show that we can obtain values of fitness or gate fidelity close to 1,
avoiding the leakage probability to higher states. The system evolution, for the gate
operation, is presented with the dynamics of the probability density, as well as a
visualization of the current of the pseudospin, characteristic of a graphene structure.
Therefore, we conclude that is possible to use the states of the graphene quantum
dot (selecting the dot size and magnetic field) to design and control the qubit subspace,
with these two time-dependent interactions, to obtain the optimal parameters for a
good gate fidelity using GA.

Background

Quantum computing (QC) has played an important role as a modern research topic because
the quantum mechanics phenomena (entanglement, superposition, projective measurement)
can be used for different purposes such as data storage, communications and data processing,
increasing security, and processing power.

The design of quantum logic gates (or quantum gates) is the basis for QC circuit model.
There have been proposals and implementations of the qubit and quantum gates for several
physical systems [1], where the qubit is represented as charge states using trapped ions, nuclear magnetic
resonance (NMR) using the magnetic spin of ions, with light polarization as qubit
or spin in solid-state nanostructures. Spin qubits in graphene nanoribbons have been
also proposed. Some obstacles are present, in every implementation, related to the
properties of the physical system like short coherence time in spin qubits and charge
qubits or null interaction between photons, which is necessary to design two-qubit
quantum logic gates. Most of the quantum algorithms have been implemented in NMR as
Shor's algorithm [2] for the factorization of numbers. Any quantum algorithm can be done by the combination
of one-qubit universal quantum logic gates like arbitrary rotations over Bloch sphere
axes (X(ϕ), Y(ϕ), and Z(ϕ)) or the Pauli gates () and two-qubit quantum gates like controlled NOT which is a genuine two-qubit quantum
gate.

The implementation of gates using graphene to make quantum dots seems appropriate
because this material is naturally low dimensional, and the isotope 12C (most common in nature) has no nuclear spin because the sum of spin particles in
the nucleus is neutralized. This property can be helpful to increase time coherence
as seen by the proposal of graphene nanoribbons (GPNs) [3] and Z-shape GPN for spin qubit [4].

In this work, we propose the implementation of three one-qubit quantum gates using
the states of a circular graphene quantum dot (QD) to define the qubit. The control
is made with pulse width modulation and coherent light which induce an oscillating
electric field. The time-dependent Schrodinger equation is solved to describe the
amplitude of being in a QD state Cj(t). Two bound states are chosen to be the computational basis |0〉 ≡ |ψ1/2 |1〉 ≡ |ψ− 1/2 〉 with j = 1/2 and j = −1/2, respectively, which form the qubit subspace. In this work, we studied the
general n-state problem with all dipolar and onsite interactions included so that
the objective is to optimize the control parameters of the time-dependent physical
interaction in order to minimize the probability of leaking out of the qubit subspace
and achieve the desired one-qubit gates successfully. The control parameters are obtained
using a genetic algorithm which finds efficiently the optimal values for the gate
implementation where the genes are: the magnitude (ϵ0) and direction (ρ) of electric field, magnitude of gate voltage (Vg0), and pulse width (τv). The fitness is defined as the gate fidelity at the measured time to obtain the
best fitness, which means the best control parameters were found to produce the desired
quantum gate. We present our findings and the evolution of the charge density and
pseudospin current in the quantum dot under the gate effect.

Methods

Graphene circular quantum dot

The nanostructure we used consists of a graphene layer grown over a semiconductor
material which introduces a constant mass term Δ [5]. This allows us to make a confinement (made with a circular electric potential of
constant radio (R)) where a homogeneous magnetic field (B) is applied perpendicular to the graphene plane in order to break the degeneracy
between Dirac's points K and K’, distinguished by the term τ = +1 and τ = −1, respectively.

The Dirac Hamiltonian with magnetic vector field in polar coordinates is given by
[6]:

(1)

where v is the Fermi velocity (106 m/s), b = eB/2, and j which is a half-odd integer is the quantum number for total angular momentum operator
Jz. We need to solve . Eigenfunctions have a pseudospinor form:

Due to the constant mass term and broken degeneracy, we obtain two independent Hilbert
spaces. Therefore, we can choose the space K for the definition of the computational basis of the qubit to implement the quantum
gates and to make the dynamic control following a genetic algorithm procedure.

The wave function in graphene can be interpreted as a pseudospinor of the sublattice
of atom type A or B. In order to visualize the physics evolution due to the gate operation,
we calculate the pseudospin current as the expectation values for Pauli matrices .

The selected states that we choose to form the computational basis for the qubit are
the energies (Ej): E1/2 = .2492 eV and E−1/2 = .2551 eV (and the corresponding radial probability distributions is shown in Figure 2a). The energy gap is E01 = E−1/2 − E1/2 = 5.838 meV. To achieve transitions between these two states with coherent light,
the wavelength required has to be , which is in the range of far-infrared lasers. Also, in controlling the magnetic
field B, it is possible to modify this energy gap. We present as a reference point the plot
for the density probability and the pseudospin current for the two-dimensional computational
basis |0〉 = |ψ1/2 (Figure 2b) and |1〉 = |ψ− 1/2 (Figure 2c), where a change of direction on pseudospin current and the creation of a hole (null
probability near r = 0) is induced when one goes from qubit 0 to1.

Figure 2.Diagram of genetic algorithm. Initial population of chromosomes randomly created; the fitness is determined for
each chromosome; parents are selected according to their fitness and reproduced by
pairs, and the product is mutated until the next generation is completed to perform
the same process until stop criterion is satisfied.

Quantum control: time-dependent potentials

First of all, we have to calculate the matrix representation of the time-dependent
interactions in the QD basis. Then, we have to use the interaction picture to obtain
the ordinary differential equation (ODE) for the time-dependent coefficient which
is the probability of being in a state of the QD at time t and finally obtaining the optimal parameter for gate operation.

Electric field: oscillating

These transitions can be induced by a laser directed to the QD carrying a wavelength
that resonates with the qubit states in order to trigger and control transitions in
the qubit subspace. We introduce an electric dipole interaction [7] using a time periodic Hamiltonian with frequency ω: Vlaser(t) = eϵ(t)r, with parameters ϵ(t) = ϵ0 cos ωt, ϵ0 = ϵ0(cos ρ, sin ρ), and r = r(cos φ, sin φ), the term ρ is the direction and ϵ0 is the magnitude of the electric field and are parameters constant in time. To determine
the matrix of dipolar transitions on the basis of the QD states, the following overlap
integrals must be calculated:

(3)

where l and j are the state indices. In Equation 3, the radial part defines the magnitude of the
matrix component, the angular part defines transition rules, and as a result, we get
a non-diagonal matrix; this indicates that transitions are only permitted between
neighbor states. The matrix components are complex numbers; ϵ0 directed in direction is a pure imaginary number and directed in is a real number.

Voltage pulse on site

This interaction can be applied as a gate voltage inside the QD. In order to modify
the electrostatic potential, we use a square pulse of width τv and magnitude Vg0. The Hamiltonian is

(4)

(5)

The matrix components in Equation 5 are diagonal, so this interaction only modifies
the energies on the site. Since the Heaviside function θ depends on r in Equation 4, the matrix components are the probability to be inside the quantum
dot which is different for each eigenstate, so this difference can introduce relative
phases inside the qubit subspace.

One-qubit quantum logic gates

Therefore, we have to solve the dynamics of QD problem in N-dimensional states involved,
where the control has to minimize the probability of leaking to states out of the
qubit subspace in order to approximate the dynamic to the ideal state to implement
correctly the one-qubit gates. The total Hamiltonian for both quantum dot and time-dependent
interactions is , where is the quantum dot part (Equation 1) and Vlaser(t) and Vgate(t) are the time control interactions given by Equations 3 and 4.

We expand the time-dependent solution in terms of the QD states (Equation 2) as. Therefore, the equations for the evolution of probability of being in state l
at time t, Cl(t), in the interaction picture, are given by:

(6)

The control problem of how to produce the gates becomes a dynamic optimization one,
where we have to find the combination of the interaction parameters that produces
the one-qubit gates (Pauli matrices). We solve it using a genetic algorithm [8] which allows us to avoid local maxima and converges in a short time over a multidimensional
space (four control parameters in our case). The steps in the GA approach are presented
in Figure 2, where the key elements that we require to define four our problem are chromosomes
and fitness.

In our model, the chromosomes in GA are the array of values {Vg0,τv,ϵ0,ρ}, where Vg0 is the voltage pulse magnitude, τv is the voltage pulse width, ϵ0 is the electric field magnitude, and ρ is the electric field direction. The fitness function, as a measure of the gate fidelity,
is a real number from 0 to 1 that we define as fitness(tmed) = | < Ψobj|Ψ(tmed) > |2 × | < Ψ0|Ψ(2tmed) > |2 where |Ψobj 〉 is the objective or ideal vector state, which is product of the gate operation
(Pauli matrix) on the initial state |Ψ0〉. Then, we evolve the dynamics to the measurement time tmed to obtain |Ψ(tmed)〉. Determination of gate fidelity results in the probability to be in the objective
vector state at tmed. Fitness involves gate fidelity at tmed and probability to be in the initial state at 2 tmed. This gives a number between 0 and 1, indicating how effective is the transformations
in taking an initial state to the objective state and back to the initial state in
twice of time (the reset phase).

The initial population of chromosomes ({Vg0,τv,ϵ0,ρ}) is randomly created, then fitness is determined for each chromosome (which implies
to have the time-dependent evolution of Cl(t) to the measurement time); parents are selected according to their fitness and reproduced
by pairs, and the product is mutated until the next generation is completed; one performs
the same process until a stop criterion is satisfied.

Results and discussion

The control dynamics were done considering N = 6 states, two of them are used as the qubit basis, so that the effect of the interaction
stays inside the qubit subspace . The gate operation is completed in a time window
that depends on ϵ0, and control parameters are defined to achieve operation inside a determined time
window. The possible values of the electric field direction ρ is set from 0 to 2π, pulse width τv domain is set from 0 to time window and the magnitude Vg0 is set from 0 to an arbitrary value. The genetic algorithm procedure is executed
for quantum gates σx and σy. The fitness reaches a value close to 1 near to 30 generations for both gates. The
optimal parameters found for quantum gate σx are Vg0= .0003685, τv = 4215.95, ϵ0 = .0000924, and ρ = .9931π. For σy are Vg0 = .0355961, τv = 326.926, ϵ0 = .0000735, and ρ = 1.5120π. For the quantum gate σz, genetic algorithm is not needed because for this case, ϵ0 = 0, so Equation 6 is an uncoupled ordinary differential equation (ODE) with specific
solution. To achieve this gate transformation in a determined time window, we can
calculate Vg0, so that the control values for this quantum gate are Vg0= .1859, τv = 5,000, ϵ0 = 0, and ρ = 0. In Figure 3, we plot the time evolution of the gate fidelity or fitness for the three gates.
We observe a good optimal convergence close to 1 at the time of measurement and reaching
again the reset phase. To see the state transition and the quantum gate effect in
the space, it is convenient to plot the density probability in the quantum dot and
the corresponding pseudospin current, where we see how the wave packet has different
time trajectory according to the gate transformation. For instance, the direction
and time of creation of the characteristic hole (null probability) in the middle of
the qubit one, which correspond more or less to an equal superposition of the qubit
zero and one (column 2 and row 2 in Figure 4, right). This process has to be different for σy because it introduces an imaginary phase in the evolution which is similar with the
change of the arrow directions in the pseudospin current. The same situation arises
for σz (result not shown), but in this case, we use as an initial state , which is similar to the plot of column 2 and row 2 in Figure 4 (left) and then to show explicitly the gate effect of introducing the minus in the
one state to reach a rotated state similar to plot of column 2 and row 2 in Figure 4 (right).

Figure 3.Time evolution of gate fidelity or fitness for the three gates. Plot of gate fidelity σx in the top side, σy in the middle, and σz in the bottom side; gate fidelity (FσI in blue where I is{x,y,z}) is the probability to be in the objective vector state;
measurement time is shown in orange.

Figure 4.Time evolution of probability density and pseudospin current for the quantum gate
σx and σy operation. Time evolution of density and current probability due to the effect of the produced
quantum gate σx in the left side and σy in the right side, initial state |Ψ0〉 = |0〉 (Figure 1b).

Conclusions

We show that with a proper selection of time-dependent interactions, one is able to
control or induce that leakage probability out of the qubit subspace in a graphene
QD to be small. We have been able to optimize the control parameters (electric field
and gate voltage) with a GA in order to keep the electron inside the qubit subspace
and produce successfully the three one-qubit gates. In our results, we appreciate
that with the genetic algorithm, one can achieve good fidelity and found that little
voltage pulses are required for σx and σy and improve gate fidelity, therefore making our proposal of the graphene QD model
for quantum gate implementation viable. Finally, in terms of physical process, the
visualization of the effects of quantum gates σx and σy is very useful, and clearly, both achieve the ideal states. The difference between
them (Figure 4) is appreciated in the different trajectories made by the wave packet and pseudospin
current during evolution due to the introduction of relative phase made by gate σy.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work presented here was carried out collaboration among all authors. FR and APG
defined the research problem. GA carried out the calculations under FR and APG's supervision.
All of them discussed the results and wrote the manuscript. All authors read and approved
the final manuscript.

Acknowledgments

The authors would like to thank DGAPA and project PAPPIT IN112012 for financial support
and sabbatical scholarship for FR and to Conacyt for the scholarship granted to GA.